- Email: [email protected]
A simple proof of the functional equation of a certain L-function
JOURNAL
OF NUMBER
THEORY
A Simple
3, 155-158 (1971)
Proof of the Functional of a Certain L-Function YOSHIYUKI
Department of Mathematics,
Equation
KITAOKA
Nagoya University, Chikusaku, Nagoya 464, Japan
Communicated by Y. Kawada Received March 15, 1970
By using character sums, Stark proved in 1968 the functional equation of the Epstein zeta function with a character. Using a method due to Weil, we give a simple proof which shows that the proof reduces to the case that the character is trivial.
First of all, we state the following basic lemma. LEMMA 1. Let a,, a, ,... and b, , b, ,... be two sequencesof complex numbers with a, , b, = O(nW)for some u > 0, andform
g(T) = 1 b,eznin+,
f(~) = C a,e2rin’, n>O
?I>0
p,(S) = 1 w-s, n>1 @@I
=
Gw”
Q)
#(s) = c
brs,
n>1
Y(s) = (27T-” T(s) #(s).
ds),
Assume c f 0, A > 0 and I > 0. Then thefollowing conditions (A) and (B) are equivalent:
(A) Q(s) + A-“lz[(ao/s) + (cb,/l - s)] is entire and bounded in every vertical strip and @D(s)= cA(~/~)-~Y(Z - s), (B) f(7) = cAz/2(AT/i)-1 g(-l/AT). Let I be a positive integer. For a real matrix L = (y i) with positive determinant and a holomorphic function h(T) on the upper half plane, we denote by h I L the function h 1L(T) = (ad - bc)z/2 (CT + d)-l h ( z: z f; ).
By definition,
h I(” J = h for a > 0.
155
KITAOKA
156
Let x be a character of conductor FYI> 1, and form
f,(4 = ,& x(4 ane2nin’,
Then we get the following Lemma 1’ immediately
from Lemma 1.
LEMMA 1’
The following
(A’) A,,,(s)
is entire and bounded in every vertical strip and
A,*,(s) (B’) f,(d
conditions (A’) and (B’) are equivalent:
= CJw)-s
= C,&
A,,,(1 - s),
cc, f 0, N > 01,
l(vnw -‘)(4.
Put now Rf
=
;
Gx
c
vmodm
S-Y)
Q: (5)
and
R, = ; CxizG,Ymodm. C
X(-Y) a (*)(m2N
-I),
where G, =
c
x(x) @ni(n/m)
xmodm
and
Then (B’) is clearly equivalent to f I R, = g ) R, . From now on, we assume that N is a positive integer and (m, N) = 1. Then for each integer y with (y, m) = 1, there exists an integer b with bylv = -1 mod m and so y( y, b) = (&, -,” belongs to p,,(N) for some n. A simple computation shows 4vlm)
(m2N
-‘)
= (N
-‘)
Ay,
b) ~u(WO (”
,).
SIMPLE
PROOF
OF
FUNCTIONAL
157
EQUATION
Since b, as well as y, runs over a set of prime representatives mod m, we see
-l j y(y)4Jym) with y(b) = y(y, b). Assuming furthermore
that
for (“, i) E r,,(N), where E is a character modulo N, and that
fc7>= ‘+? ( (N - l j CT> for some C # 0, the condition
(B’) is equivalent to -1
g/ fci” (N
1 R, - Rg( = 0.
Since CP fN
-1
j & - R,
x(-~){G,c- C,Gd--N)Y(Y))
0~ (5)
and fl Y(Y) = 44.L the condition (B’) holds if G,C = C,G,x(-N) aim by applying this result. Let Q(x) = 3 txAx = 4 f
E(m). We can attain our
aijxixj
i,i=l ("x
=
(Xl
, x2
,.-.,
x2,),
A = a 2k x 2k matrix)
be a positive definite integral quadratic form with level N and determinant D, and let P(x) be a spherical homogeneous polynomial of degree v with respect to Q(X). Put f(7) = 8(-r; Q, P) = ,zsk P(n) ezniQcnjT, g(T) = a(~; Q*, P*) = nE& P*(n) e2niQ*(n)r,
6411312-3
158
KITAOKA
where Q*(x) = (N/2) txA-lx and P*(x) = P(A-lx). results are well known by Schoeneberg [3]:
where I = k + v and (*/*) is Kronecker’s
Then the following
symbol. Thus we have
Theorem: The following assertions (A”) and (B”) hold for i-VN(k+V)/Z
c, =
+(-N)(
‘-p”“)
%Gi
(A”) 4*,( s1IS . en t’n-e and bounded in every vertical strip and (I,,,(s)
= CxN(L+U)/2-sAR,P(k
(B”) f,(T) = cxik+vgz I(m2N
+ v - s),
-l)(T).
Even when Q has an odd number of variables, we get an analogous theorem under some modifications by using [2]. These theorems were proved in [4] by Stark, where the spherical homogeneous polynomial P = 1. REFERENCES 1. E. HECKE, uber die Ekstimmung Dirichletscher Reihen durch ihre Functionalgleichung, Math. Ann. 112 (1936), 664-699. 2. W. PFETZER, Die Wirkung der Modubuhstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Varizblenzahl, Arch& Math. 4 (1953). 44g-454. 3. B. SCHOENEBERG,Das Verhalten von mehrfacher Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523. 4. H. M. STARK, L-functions and character sums for quadratic forms I, Actu Arith. 14 (1968), 5.
15 (1969), 6.
35-50.
H. M. STARK, L-functions
and character sums for quadratic forms II, Actu Arith.
307-317.
A. WEIL, uher die Bestimmung Dirichletscher ung, Math. Ann. 168 (1967), 149-156.
Reihen durch ihre Functionalgleich-
OF NUMBER
THEORY
A Simple
3, 155-158 (1971)
Proof of the Functional of a Certain L-Function YOSHIYUKI
Department of Mathematics,
Equation
KITAOKA
Nagoya University, Chikusaku, Nagoya 464, Japan
Communicated by Y. Kawada Received March 15, 1970
By using character sums, Stark proved in 1968 the functional equation of the Epstein zeta function with a character. Using a method due to Weil, we give a simple proof which shows that the proof reduces to the case that the character is trivial.
First of all, we state the following basic lemma. LEMMA 1. Let a,, a, ,... and b, , b, ,... be two sequencesof complex numbers with a, , b, = O(nW)for some u > 0, andform
g(T) = 1 b,eznin+,
f(~) = C a,e2rin’, n>O
?I>0
p,(S) = 1 w-s, n>1 @@I
=
Gw”
Q)
#(s) = c
brs,
n>1
Y(s) = (27T-” T(s) #(s).
ds),
Assume c f 0, A > 0 and I > 0. Then thefollowing conditions (A) and (B) are equivalent:
(A) Q(s) + A-“lz[(ao/s) + (cb,/l - s)] is entire and bounded in every vertical strip and @D(s)= cA(~/~)-~Y(Z - s), (B) f(7) = cAz/2(AT/i)-1 g(-l/AT). Let I be a positive integer. For a real matrix L = (y i) with positive determinant and a holomorphic function h(T) on the upper half plane, we denote by h I L the function h 1L(T) = (ad - bc)z/2 (CT + d)-l h ( z: z f; ).
By definition,
h I(” J = h for a > 0.
155
KITAOKA
156
Let x be a character of conductor FYI> 1, and form
f,(4 = ,& x(4 ane2nin’,
Then we get the following Lemma 1’ immediately
from Lemma 1.
LEMMA 1’
The following
(A’) A,,,(s)
is entire and bounded in every vertical strip and
A,*,(s) (B’) f,(d
conditions (A’) and (B’) are equivalent:
= CJw)-s
= C,&
A,,,(1 - s),
cc, f 0, N > 01,
l(vnw -‘)(4.
Put now Rf
=
;
Gx
c
vmodm
S-Y)
Q: (5)
and
R, = ; CxizG,Ymodm. C
X(-Y) a (*)(m2N
-I),
where G, =
c
x(x) @ni(n/m)
xmodm
and
Then (B’) is clearly equivalent to f I R, = g ) R, . From now on, we assume that N is a positive integer and (m, N) = 1. Then for each integer y with (y, m) = 1, there exists an integer b with bylv = -1 mod m and so y( y, b) = (&, -,” belongs to p,,(N) for some n. A simple computation shows 4vlm)
(m2N
-‘)
= (N
-‘)
Ay,
b) ~u(WO (”
,).
SIMPLE
PROOF
OF
FUNCTIONAL
157
EQUATION
Since b, as well as y, runs over a set of prime representatives mod m, we see
-l j y(y)4Jym) with y(b) = y(y, b). Assuming furthermore
that
for (“, i) E r,,(N), where E is a character modulo N, and that
fc7>= ‘+? ( (N - l j CT> for some C # 0, the condition
(B’) is equivalent to -1
g/ fci” (N
1 R, - Rg( = 0.
Since CP fN
-1
j & - R,
x(-~){G,c- C,Gd--N)Y(Y))
0~ (5)
and fl Y(Y) = 44.L the condition (B’) holds if G,C = C,G,x(-N) aim by applying this result. Let Q(x) = 3 txAx = 4 f
E(m). We can attain our
aijxixj
i,i=l ("x
=
(Xl
, x2
,.-.,
x2,),
A = a 2k x 2k matrix)
be a positive definite integral quadratic form with level N and determinant D, and let P(x) be a spherical homogeneous polynomial of degree v with respect to Q(X). Put f(7) = 8(-r; Q, P) = ,zsk P(n) ezniQcnjT, g(T) = a(~; Q*, P*) = nE& P*(n) e2niQ*(n)r,
6411312-3
158
KITAOKA
where Q*(x) = (N/2) txA-lx and P*(x) = P(A-lx). results are well known by Schoeneberg [3]:
where I = k + v and (*/*) is Kronecker’s
Then the following
symbol. Thus we have
Theorem: The following assertions (A”) and (B”) hold for i-VN(k+V)/Z
c, =
+(-N)(
‘-p”“)
%Gi
(A”) 4*,( s1IS . en t’n-e and bounded in every vertical strip and (I,,,(s)
= CxN(L+U)/2-sAR,P(k
(B”) f,(T) = cxik+vgz I(m2N
+ v - s),
-l)(T).
Even when Q has an odd number of variables, we get an analogous theorem under some modifications by using [2]. These theorems were proved in [4] by Stark, where the spherical homogeneous polynomial P = 1. REFERENCES 1. E. HECKE, uber die Ekstimmung Dirichletscher Reihen durch ihre Functionalgleichung, Math. Ann. 112 (1936), 664-699. 2. W. PFETZER, Die Wirkung der Modubuhstitutionen auf mehrfache Thetareihen zu quadratischen Formen ungerader Varizblenzahl, Arch& Math. 4 (1953). 44g-454. 3. B. SCHOENEBERG,Das Verhalten von mehrfacher Thetareihen bei Modulsubstitutionen, Math. Ann. 116 (1939), 511-523. 4. H. M. STARK, L-functions and character sums for quadratic forms I, Actu Arith. 14 (1968), 5.
15 (1969), 6.
35-50.
H. M. STARK, L-functions
and character sums for quadratic forms II, Actu Arith.
307-317.
A. WEIL, uher die Bestimmung Dirichletscher ung, Math. Ann. 168 (1967), 149-156.
Reihen durch ihre Functionalgleich-